On the Riemann Hypothesis and Hilbert's Tenth Problem

The Riemann hypothesis is one of the most famous unresolved problems in modern mathematics. The discussion here will present an overview of past methods that prove the Riemann hypothesis is a ${\mathrm{\Pi }}_1^0$ sentence. We also end with some attempts towards showing the Elliott-Halberstam conjecture is ${\mathrm{\Pi }}_1^0$.